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Chapter 4: Problem 38

Prove that a triangle can have, at most, one obtuse angle.

Short Answer

Expert verified
To prove that a triangle can have, at most, one obtuse angle, we assume that a triangle has two obtuse angles, say angle \(A\) and angle \(B\). Since both angle \(A\) and angle \(B\) are obtuse, their sum will be greater than 180 degrees. Adding angle \(C\) to the sum of angle \(A\) and angle \(B\), we have: \[A + B + C > 180掳 + C\]. However, we know that the sum of the angles in a triangle is equal to 180 degrees, which means \[A + B + C = 180掳\]. But from our calculation, we concluded that \(A + B + C > 180掳 + C\), which is a contradiction. Therefore, our assumption that a triangle can have two obtuse angles is false, and a triangle can have, at most, one obtuse angle.

Step by step solution

01

Sum of angles in a triangle

Recall that the sum of the interior angles of a triangle is always equal to 180 degrees. This will be an important fact to help us prove that a triangle cannot have more than one obtuse angle.
02

Assume a triangle has two obtuse angles

We will use proof by contradiction. Let's assume that a triangle has two obtuse angles, say angle \(A\) and angle \(B\).
03

Add the two obtuse angles

Now, let's add the two obtuse angles together. An obtuse angle is an angle greater than 90 degrees. Since we have assumed that both angle \(A\) and angle \(B\) are obtuse, their sum will be greater than 180 degrees. Mathematically, this can be represented as: \[ A + B > 180掳 \]
04

Add the third angle

Now, let's consider the third angle of the triangle, angle \(C\). It must also be greater than 0 degrees, as it is an interior angle of a triangle. Adding angle \(C\) to the sum of angle \(A\) and angle \(B\), we have: \[ A + B + C > 180掳 + C \]
05

Sum of angles in triangle contradiction

However, we know that the sum of the angles in a triangle is equal to 180 degrees. This means that: \[ A + B + C = 180掳 \] But from step 4, we concluded that \(A + B + C > 180掳 + C\), which is a contradiction. Therefore, our assumption that a triangle can have two obtuse angles is false.
06

Conclusion

After proving that a triangle cannot have two obtuse angles, it directly follows that a triangle can have, at most, one obtuse angle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Interior Angles
Understanding the sum of interior angles is fundamental when studying triangles in geometry. Every triangle has three angles, and if you add these angles together, the total will always be 180 degrees. This rule is not just a random fact; it is a cornerstone of triangular geometry and is true no matter what type of triangle you're dealing with鈥攚hether it's scalene, isosceles, or equilateral.
  • The angle sum property is essential for solving many problems involving triangles, including finding missing angles and proving potential relationships between angles.
  • One common exercise is to demonstrate that a triangle cannot have more than one obtuse angle, which relies heavily on this angle sum property.
This rule provides structure to our understanding of triangles and limits what is geometrically possible within their boundaries.
Proof by Contradiction
Proof by contradiction is a powerful mathematical tool used to establish the truth of a statement by showing that assuming the opposite leads to an inconsistency or a paradox. This approach is also known in mathematics as a reductio ad absurdum proof.
  • To apply this method, we first assume the negation of what we want to prove.
  • If this assumption leads to an impossible situation or contradicts a well-established fact, we can conclude that the original statement must be true.
In the context of geometry, this technique helps us show that certain geometric conditions are not feasible, such as the impossibility of a triangle having more than one obtuse angle, which we'll explore in relation to other key concepts here.
Obtuse Angle Definition
An obtuse angle is one of the several types of angles found in geometry, characterized by its measure being greater than 90 degrees but less than 180 degrees. Here's a quick overview:
  • An obtuse angle creates what is known as an obtuse-angled triangle when it's one of the three interior angles of a triangle.
  • Since the total sum of angles in any triangle is 180 degrees, having an obtuse angle significantly restricts the range of possible measures for the other two angles.
The concept of an obtuse angle is essential not just in triangle geometry but also in understanding more complex shapes and the behavior of angles in various contexts.
Interior Angles of a Triangle
Diving deeper into the properties of a triangle, the interior angles are those found between the sides of a triangle. Here are two essential points to remember:
  • The interior angles of a triangle always add up to 180 degrees, as established earlier.
  • Each of these angles can be acute (less than 90 degrees), right (exactly 90 degrees), or obtuse (greater than 90 degrees), but not more than one angle can be obtuse as that would violate the angle sum property of triangles.
When examining or proving properties related to triangles, this understanding is critical in determining the shape and nature of the triangle in question.
Geometry Contradiction
A geometry contradiction occurs when an assumption leads to a conclusion that defies established geometric principles or previous findings. This often signals an error in reasoning or in the initial assumption.
  • Contradictions play a crucial role in proof by contradiction; they allow us to invalidate a false supposition and thereby confirm the truth of the proposition in question.
  • In our specific case of a triangle with more than one obtuse angle, the contradiction arises when this assumption results in a total angle sum that exceeds 180 degrees鈥攕omething that is impossible for a triangle.
Recognizing and understanding contradictions helps build solid logical foundations in geometry and is crucial for mastering geometric proofs.

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Most popular questions from this chapter

(a)Let \(\mathrm{ABC}\) be a right triangle with \(\mathrm{m} \angle \mathrm{BCA}=90^{\circ}\) and \(\mathrm{m} \angle \mathrm{CAB}=30^{\circ}\). What is \(\mathrm{m} \angle \mathrm{ABC} ?\) (b) Prove that in a right triangle the sum of the measures of the angles adjacent to the hypotenuse is \(90^{\circ}\).

Prove that the base angles of an isosceles right triangle have measure \(45^{\circ}\).

Is every equilateral triangle isosceles? Is every isosceles triangle equilateral? Is every nonscalene triangle equilateral?

The measure of the vertex angle of an Isosceles triangle exceeds the measure of each base angle by \(30^{\circ}\). Find the value of each angle of the triangle.

Prove that a scalene triangle has has no 2 angles congruent.
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